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How Do I Know If My Mesh is Good Enough?

Stress Sensitivity to Mesh Density
June 6, 2014 By: Steven Hale

Here at CAE Associates, we get a lot of different questions from our clients during training classes and technical support. One of the more common questions is about finite element mesh quality.  Finite element preprocessors have come a long way over the years, to the point where users with minimal training can create meshes that appear “good”.  But, how can you really know if the mesh is good enough for your analysis?  Meshes that are "good enough" are ones that produce results with an acceptable level of accuracy, assuming that all other inputs to the model are accurate.  Mesh density is a significant metric used to control accuracy (element type and shape also affect accuracy).  Assuming no singularities are present, a high-density mesh will produce results with high accuracy.  However, if a mesh is too dense, it will require a large amount of computer memory and long run times, especially for multiple-iteration runs that are typical of nonlinear and transient analyses.

One of the ways to evaluate the quality of your mesh (and a model overall) is to compare results to test data or to theoretical values.  Unfortunately, test data and theoretical results are often not available.  So, other means of evaluating mesh quality are needed.  These include mesh refinement and interpretations of results discontinuities.

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The most fundamental and accurate method for evaluating mesh quality is to refine the mesh until a critical result, such as the maximum stress in a specific location converges (i.e. it doesn’t change significantly with each refinement).   An example is shown in Figure 1, where a 2D bracket model is constrained at its top end and subjected to a shear load at the edge on the lower right.  This generates a peak stress in the fillet, as shown.  The curve shows that as the mesh density increases, the peak stress in the fillet increases.  Ultimately, increasing the mesh density further produces only minor increases in peak stress.  In this case, an increase from 1134 elements per unit area to 4483 elements per unit area yields only a 1.5% increase in stress.

Figure 1: Stress Sensitivity to Mesh Density

The problem with this method is that it requires multiple remeshing and re-solving operations.  While this method is fine for simple models, it can be very time-consuming for complex models.  Another option is to evaluate the magnitude of stress discontinuity between adjacent elements in the critical region.  In most cases, the finite element method computes stresses directly at interior locations of the element (Gauss points) and extrapolates them to the nodes on the element boundaries.  While it is common to view these stresses as average values, the reality is that each element calculates different stresses at shared nodes, as illustrated in Figure 2.  The degree of stress discontinuity decreases with improving mesh quality, so this metric can be used to gauge mesh quality.

Figure 2: Example of Stress Discontinuities Between Adjacent Elements

Building on the earlier example, Figure 3 shows the relative difference in unaveraged stresses at shared nodes in the fillet region of the bracket.  These percentages were calculated by taking the difference in unaveraged stresses and dividing them by the nodal averaged stress.  The finer mesh shown on the right generates much lower relative differences in the fillet which indicates that this mesh is considerably more accurate.  The percentage difference also indicates the degree of potential error in the solution.  While other error measures can be used, they are generally all based on the difference in the critical results between adjacent elements at their shared nodes.  

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It should be noted that it is quite common and perfectly acceptable to have high relative stress differences in regions farther from the critical locations in your model where stresses are lower.  This is often the case because these regions are not meshed at a high density and sometimes because they contain singularities.  However, it is up to the analyst to determine if a high degree of accuracy is important in a given region and, if it is, to evaluate the quality of the mesh in that region.  Mesh quality is extremely important to overall model accuracy for your FEA consulting projects and can ultimately mean the difference between predicting that a design will or will not fail.

Figure 3: Relative Difference in Stresses at Shared Nodes for a Coarse Mesh (L) & a Finer Mesh (R)